Litz Wire Calculator: Strand Count, Diameter and AC Resistance

For copper, $\rho(T)=\rho_{20}[1+\alpha(T-20)]$ and $\delta_h=\sqrt{\rho/(\pi h f\mu_0)}$. The copper base resistivity follows Sullivan–Zhang (2014), attached p. 2. The 20 °C coefficient 0.00393 K⁻¹ follows NBS Handbook 100, Table 3. All currents entered are RMS. The straight conductor length is total series turns times mean turn length; litz adds the specified strand take-up.

Strand losses. Sullivan–Zhang (2014), attached pp. 3–5, gives the fine-strand external winding-field term

Round-strand transport skin loss is its low-frequency expansion, $F_s=1+d^4/(768\delta^4)$. The coefficient follows Peck (1992), printed p. 26, Eq. (13); at $d=\delta$ this expansion gives 1.001302 versus the exact cylindrical value 1.001301.

Bundle self-field correction, derived here. In a uniformly populated circular bundle, $B(r)=\mu_0 I r/(2\pi R^2)$, hence $\langle B^2\rangle=\mu_0^2I^2/(8\pi^2R^2)$. Integrating the fine-strand eddy-current loss $\pi\ell d^4\langle\dot B^2\rangle/(64\rho)$ gives the added strand-level proximity factor $F_i=n^2d^6/(128\delta^4D^2)$, with $D^2=n(d+2t)^2/k_b$. This continuum correction describes field variation inside a transposed bundle; it does not estimate circulating currents between badly transposed strands. The external winding-field term describes the bundle-centre field. Their cross term averages to zero under the uniform circular-bundle assumption. Total ideal-litz factor is $F_s+F_e+F_i$.

Optimization. The enamelled-circle constraint is $Nn\pi(d+2t)^2/4\leq k_wbh$. A round bundle must also satisfy $D=\sqrt{n/k_b}(d+2t)\leq\min(b,h)$. The candidate count cap is the smaller of the area limit and $\lfloor k_b[\min(b,h)/(d+2t)]^2\rfloor$. Individual enamelled strands and the insulated solid-round comparator are screened against both window dimensions. For fixed diameter, harmonic loss has the form $A/n+Bn+C$. Check adjacent integers near $\sqrt{A/B}$ for each allowed outer-stage product, constrained by window area, maximum count and first-bunch size. Compare all 121 diameters. The optimizer minimizes copper loss; no price model is used.

Deriving the simplified design constant. Solving $F_e=F_{\rm target}-1$ gives $k=\sqrt{192(F_{\rm target}-1)}/(\pi d^3)$ and $n_e=k\delta_{\rm eff}^2b/N_s$. The factor is selected by the user. No values from the paper's economical-stranding table are copied. This separate count guide excludes the added self-field and skin terms.

Harmonics. $P=I_{\rm dc}^2R_{\rm dc}+\sum_{h=1,3,5}I_h^2R_{\rm ac}(hf)$. The effective frequency is $f_{\rm eff}=f\sqrt{\sum h^2I_h^2/\sum I_h^2}$, including DC in the denominator.

Solid round and foil. The round conductor uses the cylindrical diffusion solution. With $x=a/\delta$, $z=(1+j)x$, and $S=zI_0(z)/(2I_1(z))$, transport factor is $\Re(S)$ and transverse-field shielding multiplies the fine-strand external term by $-4\Im(1/S)/x^2$. A convergent continued fraction evaluates the Bessel ratio; a large-$x$ expansion prevents numerical overflow. This is a uniform-field estimate, not an exact packed-winding solution. Full-breadth foil uses Dowell's whole-section factor, with $u=t_f/\delta$:

Small- and large-argument limits avoid cancellation and overflow.

• Copper, round individually insulated strands, constant copper temperature, equal strand currents and adequate transposition; negligible displacement current.
• Overall winding field is one-dimensional, parallel to layer faces and proportional to the entered winding current at every harmonic. $N_s$ represents identical series MMF sections. Different winding-current waveforms require a resistance-matrix model.
• Enamel thickness, winding packing, bundle packing and take-up are user inputs, held constant across the search. Serving, filler and extra mechanical margins must be allowed for in the available build or packing input.
• The bundle self-field correction is a continuum approximation derived in this calculator, not an additional empirical coefficient from Sullivan–Zhang. Local packing-field perturbations are neglected.
• The solid comparison uses an equal-copper cylinder and an averaged winding field. Foil is full-breadth, with one turn per layer. Their DC lengths exclude litz twist take-up.

The litz optimizer is restricted to $d\leq\delta$ at the highest nonzero entered harmonic. It finds the best of the sampled diameters and all permitted equal-group integer constructions within the count cap. It is not a continuous global optimum or a catalogue availability guarantee. Search-range errors apply only in optimization mode. An unavailable search produces an empty optimization sweep, while valid manual-wire results remain available.

The first-operation limit $n_{1,\max}=4(\delta/d)^2$ and subsequent 3-, 4- or 5-way stages follow Sullivan–Zhang (2014), attached pp. 3 and 5. The manual-mode frequency screen uses the entered first-bunch count. Bundle-level proximity currents require twist pitch short compared with the length exposed to a field, and radial transposition is needed to prevent bundle-level skin redistribution. These excess losses cannot be inferred from diameter and count alone. If a construction fails the screen, displayed losses remain ideal-litz estimates and actual losses can be higher. Unequal-group or unverified manual constructions need manufacturer confirmation. The suggested recipe is an admissible alternative for the evaluated count; the manual first-bunch entry is screened separately.

The geometric screen checks enamelled-circle area plus necessary round-envelope dimensions. Comparison utilization is the larger of the relevant area and dimension ratios; foil uses stack build divided by available build. A value no greater than one is necessary, not a fit guarantee. Finished diameter, serving, leads and bend radii require a winding layout. The foil fit check assumes the allocated build is available across the same breadth. The frequency sweep holds construction fixed, uses sinusoidal current at each point and omits the litz curve above the lower of the fine-strand and first-bunch screening frequencies. Both screens and the resulting curve limit are shown beside the chart.

Round-wire proximity estimates become uncertain when conductors are tightly packed and diameter exceeds skin depth. Foil assumes a 1-D field with negligible edge effects. No numerical precision displayed here represents a measured accuracy guarantee.

Not covered: air-gap fringing, isolated spiral/solenoid field geometry, arbitrary harmonics beyond those entered, unequal parallel paths, termination loss, core loss, dielectric loss, self-resonance, temperature rise or allowable current rating.